Name Class Date. t = = 10m. n + 19 = = 2f + 9

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1 1-4 Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equality properties of real numbers and inverse operations to rewrite the equation until the variable is alone on one side of the equation. Whatever remains on the other side of the equation is the solution. Addition Property of Equality To isolate w on one side of the equation, add 7 to each side. w w 18 Multiplication Property of Equality To isolate w on one side of the equation, multiply each side by. w 18 w = 36 Subtraction Property of Equality To isolate z on one side of the equation, subtract 3 from each side. z z 10 Division Property of Equality To isolate z on one side of the equation, divide each side by. z 10 z Solve each equation. 1. y + 1 = 8. p 9 = r = = 1 + k. 9q = t = 4 7. = 7 d = 10m 9. 3g 14 = n + 19 = = 4 c 1. 1 = f + 9 Prentice Hall Algebra Teaching Resources 39

2 1-4 Reteaching (continued) Solving Equations To solve an equation for one of its variables, rewrite the equation as an equivalent equation with the specified variable on one side of the equation by itself and an expression not containing that variable on the other side. ax The equation b x b defines a relationship between a, b, and x. What is x in terms of a and b? Use the properties of equality and the properties of real numbers to rewrite the equation as a sequence of equivalent equations. ax b x b ax b ( x b) ax b ( x b) Simplify. ax - b = x + 4b ax - x = 4b + b ax - x = b x(a - ) = b Multiply each side by. Distributive Property Add and subtract to get terms with x on one side and terms without x on the other side. Simplify. Distributive Property b x Divide each side by a. a The final form of the equation has x on the left side by itself and an expression not containing x on the right side. Solve each equation for the indicated variable m n = m + n, for m 14. (u + 3v) = w u, for u 1. ax + b = cx + d, for x 16. k(y + 3z) = 4(y - ), for y r + 3s = 1, for r 18. f g 1 fg,for f 3 1 x k 3 a 3y 19., for x 0. 4 a y, for y j 4 b Prentice Hall Algebra Teaching Resources 40

3 1- Reteaching Solving Inequalities As with an equation, the solutions of an inequality are numbers that make it true. The procedure for solving a linear inequality is much like the one for solving linear equations. To isolate the variable on one side of the inequality, perform the same algebraic operation on each side of the inequality symbol. The Addition and Subtraction Properties of Inequality state that adding or subtracting the same number from both sides of the inequality does not change the inequality. If a < b, then a + c < b + c. If a < b, then a c 0, then ac < bc. If a 0, then a c < b c. What is the solution of 3(x + ) 1 x? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. 3x x 3x x 4x x 0 x Distributive Property Simplify. Addition Property of Inequality Subtraction Property of Inequality Division Property of Inequality To graph the solution, locate the boundary point. Plot a point at x =. Because the inequality is less than or equal to, the boundary point is part of the solution set. Therefore, use a closed dot to graph the boundary point. Shade the number line to the left of the boundary point because the inequality is less than. Graph the solution on a number line. Solve each inequality. Graph the solution. 1. x + 4(x ) > 4. 4 (x 4) (4x + 3) Prentice Hall Algebra Teaching Resources 49

4 1- Reteaching (continued) Solving Inequalities The procedure for solving an inequality is similar to the procedure for solving an equation but with one important exception. The Multiplication and Division Properties of Equality also state that, when you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol. If a bc. If a < b and c < 0, then a b. c c What is the solution of x 3(x 1) < x +? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. x 3(x 1) < x + x 3x + 3 < x + Distributive Property x + 3 < x + Simplify. x < Subtraction Property of Inequality x > 1 Division Property of Inequality The direction of the inequality changed in the last step because we divided both sides of the inequality by a negative number. Graph the solution on a number line. Solve each inequality. 3. x 1 4( x) (x 7) > 4 + x. 7(x + 4) (3 + x) 6. 4x 1 < 6x Prentice Hall Algebra Teaching Resources 0

5 1-6 Reteaching Solving absolute value equations require solving two equations separately. Recall that for a real number x, x is the distance from zero to x on the number line. The equation x = p means that either x = p or x = -p because both are p units from 0. Absolute Value Equations and Inequalities What is the solution set for the equation x 1 3 = 4? The first step in solving an absolute value equation is to isolate the absolute value on one side of the equal sign. x 1 3 = 4 x = Add 3 to each side. x 1= 7 Simplify. Next, rewrite the absolute value as two equations and solve each of them separately. x + 1 = 7 or x + 1 = 7 Definition of absolute value x = 6 or x = 8 Addition Property of Equality x = 6 8 or x = Division Property of Equality Notice that the same operations are performed in the same order on each of the two equations. However, do not try to simplify the process by solving a single equation. This leads to errors. The solutions are x = 6 or x = 8. Check each solution in the original equation: Check Solve each absolute value equation. Check your work. 1. x 3 4 = 3. 3x = 13 Prentice Hall Algebra Teaching Resources 9

6 1-6 Reteaching (continued) Absolute Value Equations and Inequalities To solve an absolute value inequality, keep in mind that x is the distance from zero to x on the number line. So, if x < p, then x is less than p units from 0, so x p, then x is greater than p units from 0, so x > p x p. In this case, we need to rewrite the absolute value inequality as two separate inequalities. Do not try to combine them into one inequality. What is the solution set for the inequality x 3 11? Because the inequality is >, use x > p x p. Begin by rewriting the absolute value as two equations and solve each of them separately. x + 3 < 11 or x+3>11 Rewrite as a compound inequality. x < 14 or x > 8 Subtract 3 from each side. x > 7 or x > 4 Divide each side by. The solution set is x < 7 or x > 4. x Complete the steps to solve the inequality 4 3. x 3. 4 Rewrite as a compound inequality. 4. x Add to each part.. x Multiply each part by. 6. What is the solution? Prentice Hall Algebra Teaching Resources 60

Chapter 2 Linear Equations and Inequalities in One Variable

Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound